Forward modelling 3-D geophysical electromagnetic field data with meshfree methods

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Forward modelling 3-D geophysical

electromagnetic field data with meshfree methods

by

© Jianbo Long

A thesis submitted to the School of Graduate Studies

in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Geophysics)

Department of Earth Sciences Memorial University of Newfoundland

May, 2020

St. John’s Newfoundland and Labrador

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Abstract

Simulating geophysical electromagnetic (EM) data over real-life conductivity mod-

els requires numerical algorithms that can incorporate realistically complex geometry

and topography. The most successful way to incorporate them is to use unstruc-

tured meshes in the discretization of an Earth model. Current mesh-based numerical

methods that are capable of using such meshes have inherent drawbacks caused by

generating 3-D unstructured meshes conforming to irregular geometries. Such a mesh

generation process may become computationally expensive and unstable, and partic-

ularly so for EM inversion computations in which the forward modelling may be

required many times. In this thesis I investigate the feasibility and applicability of

radial basis function-based finite difference (RBF-FD), a meshfree method, in forward

modelling 3-D EM data. In the meshfree method, the physical model is represented

using only a set of unconnected points, effectively overcoming the issues related to

the mesh generation. To improve numerical efficiency, unstructured point sets are

used in the computation for the first time for EM problems. The computation is

further accelerated by introducing a new type of radial basis function in the RBF-

FD method. The convergence and accuracy of the proposed RBF-FD method are

demonstrated first via forward modelling gravity and gravity gradient data. The

computational efficiency of the meshfree method is compared with that of using a

more traditional finite element method. The meshfree method is then applied to

forward model magnetotelluric data of which the effectiveness is demonstrated using

three benchmark conductivity models from the literature. Faithful reproduction of

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the physics in the EM fields, e.g. discontinuous electric fields across the conductivity

contrasts, is achieved by proposing a hybrid meshfree scheme which is a modifica-

tion to standard meshfree algorithms. The hybrid method is also applied to simulate

controlled-source EM data in the frame of both total-field and primary-secondary field

approaches, in which the problems in dealing with singular source functions that cause

singularities in the EM fields are addressed. For these two approaches, the accuracies

of the proposed hybrid meshfree method in forward modelling the controlled-source

EM data are demonstrated by using idealized 1-D layered models and a 3-D marine

canonical disk model. The successful applications of the proposed meshfree method

in modelling the above EM data suggest that the meshfree technique has the poten-

tial of becoming an important numerical method for simulating EM responses over

complicated conductivity models.

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Acknowledgements

Throughout the years in my PhD study, I have had great helps from many people.

First, I would like to give thanks, in a countless form indeed, to my supervisor Colin Farquharson. His spirit of endurance in guiding and supporting me throughout my research adventure is truly remarkable. Aside from generous financial supports, he is always encouraging and trying to provide ideas when I feel the road ahead is full of clouds. Quite honestly, I have never had a mentor before like him, who is doing the geophysical science at the frontiers and yet is always very humble, particularly so when he is trying to tell me something I don’t know. I am sure I will forever benefit from what I’ve learned from his spirit of doing research. Without his support, my PhD research adventure will be much more bumpy.

Specific thanks are given to my supervisory committee, Alison Malcolm and Charles Hurich (Chuck). I am very grateful for their kindness, help and dedicated efforts in co-supervising my study. Whenever I go to them for help, even just for a chat, they are always there and willing to provide support.

I also want to express my gratitude to the following people in Colin’s research

group, Masoud Ansari, Hormoz Jahandari, Angela Carter-McAuslan, Mehrdad Dar-

ijani, Xushan Lu and Peter Leli` evre. I have always enjoyed discussions (not entirely

about EM though) with them. Over the course of scientific research, we also have

formed an informal tennis team, which is extremely important to help adapt to the

not-always-beautiful weather of the lovely St John’s.

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Special appreciation goes to my family and my fianc´ ee Grace. Without their con- ditionless love and support, I wouldn’t be able to make the first step of my adventure in geophysical science, not to mention to make it this far.

Last but not least, I would like to thank our department as a whole. Their kindness

and efforts in supporting graduate students have impressed me so much. I appreciate

their tolerance even when I ask for unusual needs during my study program. I feel so

lucky to be part of the department.

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List of Figures

1.1 Two different tetrahedral meshes for the same multi-layer marine hy- drocarbon reservoir model. The thin hydrocarbon layer is shown in red in both meshes. The mesh shown in (a) has about 1 million more cells than the mesh in (b), with the latter mesh being generated with better edge length constraints. From Fig 7 in Dunham et al. (2018). . 11 1.2 Schematic illustration of model discretizations for 2-D irregular ge-

ometries using (a) rectilinear mesh with local quadtree refinements, (b) unstructured triangular mesh and (c) meshfree points. . . . . 13 1.3 Different spatial discretizations of an Earth model in mesh-based and

meshfree numerical methods: (a) a mesh describing the example Earth

model which is shown as the cross section of a conductor (in red) em-

bedded in the half space (in grey); (b) an example mesh (unstructured

tetrahedra) used for solving the equations if using mesh-based meth-

ods; (c) a cloud of points used for solving the equations when using

meshfree methods. . . . . 14

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2.1 Schematic illustration of meshfree subdomains consisting of scattered points. A set of local points within a neighbourhood (outlined by dotted circles) of a point (denoted as support point) comprises the subdomain for that point. . . . 19 2.2 Schematic illustration of different discretizations in mesh-based and

meshfree approximation methods. Shown above are (a) a grid-based finite difference stencil, (b) an unstructured mesh and (c) two meshfree subdomains. . . . 20 2.3 Some point arrangements in a 2-D meshfree subdomain: (a) regular,

equidistant points; (b) randomly distributed points and (c) collinear points. The red point is the support point. . . . 21 3.1 Quasi-uniform points shown on the surface and at the cross-section

y = 0 of the prismatic density model for convergence analysis. The red points in the middle of the cross section are inside the cubic density source. . . . 48 3.2 RMS errors of computed gravity potentials versus the averaged in-

ternodal distance h. Results were calculated by PHS RBF-FD with

enriched linear polynomials (labelled as ‘PHS+poly4’) and quadratic

polynomials (labelled as ‘PHS+poly10’). The two black solid lines in-

dicate theoretical linear O(h) and quadratic O(h

²

) convergence rates

in the log-log plot. . . . . 49

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3.3 Unstructured, non-uniform point discretization displayed in: (a) a per- spective view of the points within the easting −500 ⩽ x ⩽ 0 km, and (b) an enlarged view of the local point refinements within the blocky density source (red points, internodal distance h = 10 m) and at the observation sites at the cross section northing y = 0 m (internodal distance h = 1 m). . . . . 52 3.4 Relative errors for the numerical potential and gravity by RBF-FD

along the test line in the blocky density model (Fig 3.3). Panels a and b show the errors under different refining scales inside the density source (indicated by the internodal distance h = 40, 20, 10 and 5 m) for potentials and gravities (g

z

), respectively. In the case of source refinements, a refinement of h = 1 m was applied at the measurements.

Panels c and d show the corresponding errors under different local refining scales at the measurement sites (indicated by the internodal distance h = 20, 10, and 5 m). The situation without local refinement at the measurements is denoted as NR. In the case of measurement refinements, a refinement of h = 10 m was applied for the density source block. . . . 54 3.5 Computed potential (ϕ), vertical gravity (g

z

), and their relative errors

(Rerr) by the PHS RBF-FD using a quasi-uniform point discretization

with the averaged inter-nodal distance h = 20 m (panels a, c and e),

and using a non-uniform, unstructured point discretization (panels b,

d and f ), which is discussed in the text, for the blocky density model. 55

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3.6 Comparison of six components of gravity gradient tensor (U

_xx

, U

_xy

, U

_xz

, U

_yy

, U

_yz

, and U

_zz

) calculated using the PHS RBF-FD method with a non-uniform, unstructured point discretization, and using a summation method (Okabe, 1979) for the blocky density model. The gradient values are shown in E¨ otv¨ os (E). . . . 56 3.7 Plots of RMS errors (panel a) of potentials of PHS RBF-FD, and the

condition numbers (κ, panel b) of the local interpolation linear system for the point at (x, y, z) = (50.0, 50.0, 50.0) versus various stencil sizes, n. In the labels, ‘4’ indicates that the PHS RBF-FD was enriched with linear polynomials, and ‘10’ indicates that quadratic polynomials were instead used. ‘scaled’ indicates that the PHS RBF-based interpolation was scaled by the distance h

sp

, which is the radius of the smallest sphere enclosing all n points in a subdomain. ‘unscaled’ indicates that the interpolation was not scaled. . . . 58 3.8 Perspective 3-D views of the Bay du Nord density model. The easting

is along the x direction, and the northing is along the y direction. The top panel (a) shows the reservoir unit overlain by the sedimentary unit.

The view is looking from south to north. The bottom panel (b) is a

side view from north to south. A 2-D cross section at x = 0 (easting)

that cuts through the two blocks is given in Fig 3.9. . . . 61

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3.9 Vertical cross section of the Bay du Nord model at x = 0 (easting, looking from the positive x direction). The northing direction is along the y axis. Two vertical white lines mark the positions of two measure- ment lines at y = −3.5 km (L1) and y = −1.0 km (L2), respectively.

. . . . 62 3.10 Unstructured mesh (panel a) and meshfree points (panel b) for the

section shown in Fig 3.9. Note that the mesh is located on the 3-D crinkled surface made at easting x = 0 m, and the points in the panel (b) are the 3-D points within the rectangular domain: Ω

p

= {(x, y, z)|−

0.75 < x < 0.75 km, −6.0 < y < 1.0 km, −5.0 < z < −2.0 km}. Panel (c) shows an enlarged 3-D view of the mesh around the wedge-like attachment that is shown in the panel (a). Panel (d) shows the same enlarged view as in panel (c) but only with the tetrahedral elements in the background region. In the panel (d), the tetrahedra shown at the southern end become increasingly thin wedge- or sliver-like elements. 63 3.11 Plots of computed g

z

along the two vertical lines of the Bay du Nord

model, L1 (y = −3.5 km) and L2 (y = −1.0 km), for three different model discretizations using PHS RBF-FD (panels a and c) and scalar FE (panels b and d) methods. Analytical solutions are denoted by

‘Okabe’. The three model discretizations are distinguished by the total

number of points, N . . . . 64

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3.12 Plots of the differences between various numerical solutions and an- alytical solution of g

_z

for the Bay du Nord model. The numerical solutions, as shown in Fig 3.11, were computed along the two vertical lines, L1 (y = −3.5 km) and L2 (y = −1.0 km), for three different model discretizations using PHS RBF-FD (panels a and c) and scalar FE (panels b and d) methods. The three model discretizations are distinguished by the total number of points, N . . . . 65 4.1 Schematic illustration of different meshfree subdomains for different

points. Points on the boundaries of the problem domain and on the interfaces are coloured blue, and interior points are coloured black. . 84 4.2 Different meshfree subdomains in the partitioned regions. Shown above

are two different physical regions with the resistivities ρ

₁

and ρ

₂

. Mesh-

free points are distributed in both regions and at their interface. Sub-

domains for those support points far away from the interface (e.g., A

and D) include only the points belonging to the same region as the

support points. Subdomains for those support points close to the in-

terface (e.g., B and C) include some interfacial points as well as interior

points, but not any points on the other side of the interface. . . . 85

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4.3 Demonstration of the incorporation of the weak-form integration in the hybrid RBF-FD method. There are three steps in forming the local subdomain for an interfacial support point: (i) construct a normal meshfree subdomain as if there is no interface (panel 1); (ii) construct a local mesh among the points within the subdomain (panels 2 and 3);

and (iii) select the required sub-mesh from the mesh formed in Step 2 by excluding unnecessary points, depending on the weak form method (panel 4). . . . 88 4.4 Plan (a) and section (b) views of the COMMEMI 3D-1A conductivity

model. The conductor is indicated by the shaded block. Two perpen- dicular measurement lines are located at the air-Earth surface (z = 0 km) and are denoted as dotted lines in the plan view. The x-directed measurement sites are indicated by triangles in the section view. . . . 91 4.5 Computed impedance components Z

xy

and Z

yx

at the frequency of 0.1

Hz for line 1 (panel a, b) and line 2 (panel c, d) using the meshfree method, compared with a finite volume solution (Jahandari, 2015) and other mesh-based solutions (error bars, Zhdanov et al., 1997). For each panel, the top shows the apparent resistivity, and the bottom shows the phase curves. The error bars represent the standard deviations and mean values of the numerical results reported in Zhdanov et al. (1997).

Only the error bars for apparent resistivity are available. . . . 93

4.6 The same plots as Fig 4.5, but for the case of 10 Hz. . . . . 94

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4.7 A 2-D section view of the COMMEMI 3D-1A model at y = 0. The solid cyan line is along the x direction (−1.2 ⩽ x ⩽ 1.2 km, z = −1.25 km), and intersects the conductive block (in red) at x = −0.5 km and x = 0.5 km. . . . 95 4.8 Continuity plots of the components A

x

, ψ, (∇ψ)

_x

and E

x

for the hor-

izontal line shown in Fig 4.7. The left column shows the plots of the real part of the four quantities, and the right column shows the imag- inary part of them. The four components were calculated in the E-x polarization mode with f = 0.1 Hz using the mixed RBF-FD meshfree method. All values are plotted in the log

10

scale. Unfilled black circles indicate that the values are negative, and filled blue circles represent positive values. . . . 96 4.9 The same continuity plots as Fig 4.8 but with the impedance results

calculated using the continuous version of the RBF-FD meshfree method. 97 4.10 The same plots of the off-diagonal impedance components as Fig 4.5

but with the impedance results calculated using the continuous version

of the RBF-FD meshfree method. . . . 98

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4.11 2-D vector arrow plots of the E field in the horizontal plane z = −500 m for the COMMEMI 3D-1A model. The results were calculated us- ing the mixed RBF-FD meshfree method for the frequency f = 0.1 Hz (the plots for f = 10 Hz are similar). The “Ex mode” in the labels means the external electric field of the MT source is parallel to the x direction (i.e., E-x polarization), and “Ey mode” means the E-y polar- ization mode. Both real (denoted by “real”) and imaginary (denoted by “imag”) parts of the field are shown here. The magnitude of the E field is indicated by the colour bar. . . . 100 4.12 The same 2-D vector arrow plots as Fig 4.11 but with the results cal-

culated using the continuous version of the RBF-FD meshfree method. 101 4.13 Coloured images of computed ∇ · A, ∇ · (iωA) and ∇ · (∇ψ) in the

horizontal plane z = −500 m for the COMMEMI 3D-1A model. The frequency is 0.1 Hz. The source polarization mode is E-x polarization (the results for the E

y

polarization mode are similar). Panels (a)-(f ) show the divergence results using the first discretization in Table 4.2 (hence ‘Discret-1’) with 7,414 points. Similarly, panels (h)-(m) show the results using the 4th discretization in Table 4.2 (‘Discret-4’) with 58,044 points. The magnitudes of the values are indicated by the colour bar. The highest value in the colour bar is -3.6. . . . 103 4.14 The same coloured image plots as in Fig 4.13, but for the frequency of

10 Hz. The highest value in the colour bar is -2.8. . . . 104

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4.15 Plan (a) and section (b) views of the COMMEMI 3D-2A conductivity model. The subsurface (z < 0) has three layers with different con- ductivities as shown in the panel (b). In the first layer (the top layer, σ = 0.1 S/m), there are one rectangular conductor (σ = 1 S/m) and one intrusion block from the second more resistive layer (σ = 0.01 S/m). The red dotted line at y = 0 (plan view panel) represents the measurement sites at the air-Earth surface (z = 0) used to calculate the MT response for the model. . . . 106 4.16 A 3-D perspective view of the meshfree point discretization of the

COMMEMI 3D-2A model. The points are within the domain Ω = {(x, y, z)| − 80 ⩽ x ⩽ 80 km, −5 ⩽ y ⩽ 5 km, −120 ⩽ z ⩽ 30 km}. . . 107 4.17 Computed Z

xy

(panel a) and Z

yx

(panel b) components of the impedance

tensor for the COMMEMI model 3D-2A using the hybrid meshfree

method. The frequency is f = 0.1 Hz. Also plotted are a FV so-

lution (Jahandari, 2015) and an IE solution (Wannamaker, 1991) for

comparison. . . . 107

4.18 The same plots as in Fig 4.17, but for the case of f = 0.01 Hz. . . . . 108

4.19 The same plots as in Fig 4.17, but for the case of f = 0.001 Hz. . . . 108

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4.20 Section (a) and plan (b) views of the Dublin Test Model 1 (DTM-1).

There are three rectangular prisms attached together in the subsurface with different resistivities (ρ

₁

, ρ

₂

, ρ

₃

) as shown above. The background subsurface has the conductivity of σ

_b

= 0.01 S/m (ρ

_b

= 100 Ωm). The origin of the coordinate system is at the surface and marked as the red X in the plan view. The four blue dotted lines in the plan view are the designed measurement lines at the air-Earth surface. . . . . 110 4.21 Perspective 3-D views of the DTM-1 model at the section x = 0. Panels

(a) and (b) show the plane of the cross section and the three blocky targets at two different angles. Panel (c) shows the meshfree point discretization (−10 ⩽ x ⩽ 10 km) that was used in the meshfree calculation. The air-Earth surface is at z = 0. . . . 110 4.22 Computed meshfree results of impedance components compared with

various other numerical solutions (Miensopust et al., 2013) for the pe- riod range T ∈ [0.1, 10000] sec at the origin site (see Fig 4.20) for the DTM-1 model. The first column of the curves shows the solutions cor- responding to the first column of the User/algorithm list at the bottom;

the same corresponding relation also applies to the second and third

columns of the curves. . . . 111

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5.1 Schematic illustration of the model separation in a primary-secondary field approach: (a) an original inhomogeneous conductivity model with EM transmitter (TX) and receiver (RX), (b) a 1-D half space model that is chosen as the background model and (c) the anomalous conduc- tivity distribution in which σ

anomaly

= σ

original

− σ

background

as a result of the choice of the background model. The anomalous conductivity distribution acts as the source term when numerically solving for the scattered secondary EM field (see details in Appendix E) . . . . 115 5.2 A diagram illustrating (a) a horizontal electric dipole source (the in-

duced electric currents in the subsurface are represented as thin lines with arrow) and (b) a magnetic loop source. . . . 118 5.3 Different situations of E field discontinuity: (a) E is discontinuous

at the ends of an electric dipole source in all directions and (b) E is discontinuous in the normal direction at the conductivity interfaces but continuous in the tangential direction in the case of σ

1

̸= σ

2

. In the diagram, the length of an arrow represents the magnitude of E field. . 120 5.4 Schematic illustration of dealing with electric dipole source in the hy-

brid RBF-FD method. Shown above is a local mesh for the source

point where EM fields are expected to be discontinuous. . . . . 120

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5.5 Comparison of the calculated E

_x

responses for the whole space model (σ = 1 S/m) using the total-field FE approach and the analytical approach. Positive and negative values are indicated by “+” and “-”, respectively, in the figure’s legends in addition to different colours of the symbols. Real and imaginary components of E

x

are denoted by different symbols. The results are for three different frequencies: (a) 1 Hz, (b) 0.1 Hz and (c) 10

⁻⁴

Hz. . . . 124 5.6 Same comparison plots as in Fig 5.5, but for shorter offsets from the

source. . . . 126 5.7 Vertical section of the unstructured tetrahedral mesh used for the half

space model. . . . 127 5.8 Total-field FE result of E

x

component for the half space model com-

pared with analytical solution. The source is a horizontal electric dipole. Positive values are indicated by coloured symbols, and neg- ative values are indicated by black symbols. The frequency is f = 1 Hz. . . . 128 5.9 Hybrid meshfree (MM) numerical result of E

x

component for the half

space model compared with the total-field FE result. The frequency is

f = 1 Hz. Minimum FE treatments for source points (SP) were used

in the meshfree solution. . . . 129

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5.10 Hybrid meshfree numerical result of E

_x

component for the half space model compared with the total-field FE result. The frequency is f = 1 Hz. In the meshfree solution, FE treatments for source points were applied for any points within R

_s

= max{|x|, |y|, |z|} = 150 m in the meshfree point discretization. . . . . 130 5.11 Same plots as in Fig 5.10, but with R

s

= 1 km in the hybrid meshfree

method. . . . 131 5.12 Same plots as in Fig 5.10, but with R

s

= 3 km in the hybrid meshfree

method. . . . 132 5.13 Same plots as in Fig 5.12, but over extended offsets and using more

frequencies: (a) 1 Hz, (b) 0.1 Hz and (c) 10

⁻⁴

Hz. . . . 133 5.14 Hybrid meshfree (MM) numerical result of E

x

component for the half

space model compared with the total-field FE result. The 200 m long dipole source is located at z = −500 m. The frequency is f = 1 Hz. . 134 5.15 Diagram of the marine hydrocarbon disk model. The disk is located

in the sea bed layer. . . . . 137 5.16 Comparison of the calculated E

x

responses for the canonical disk model

using the primary-secondary (PS) FE approach and total-field FE ap- proach. In both implementations, the linear systems of equations were solved using a direct solver (“DR”). Positive and negative values are indicated by “+” and “-”, respectively, in the figure in addition to different colours of the symbols. The frequency is f = 1 Hz. . . . . . 138 5.17 Same comparison plots as in Fig 5.16, but for the frequency f = 10

⁻⁴

Hz. . . . 139

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5.18 Scatter plot of the primary-secondary (PS) scalar FE solution of E

_x

in a comparison with a total-field vector FE solution (Ansari & Far- quharson, 2014) for the frequency f = 1 Hz. Symbols of the vector FE solution are indicated as “vFE”. . . . 140 5.19 Scatter plot of the primary-secondary (PS) meshfree (“MM”) solution

of E

x

in a comparison with a total-field vector FE (“vFE”) solution (Ansari & Farquharson, 2014) for the frequency f = 1 Hz. . . . . 141 5.20 Comparison plots of the primary-secondary (PS) meshfree (“MM”)

solution and PS FE solution of E

x

in the canonical disk model for three different frequencies f = 0.1, 10

⁻⁴

and 10

⁻⁸

Hz. . . . 142 D.1 Computed impedance components Z

_xy

and Z

_yx

at the frequency of 0.1

Hz for line 1 (panel a, b) and line 2 (panel c, d) using the hybrid meshfree method for the 1st discretization (total number of points equal to 7,414) in Table 4.2. Also shown are a finite volume solution and other mesh-based solutions (see Fig 4.5). . . . 179 D.2 The same plots as in Fig D.1 but using the 4th discretization (total

number of points equal to 58,044) in Table 4.2. . . . 180 F.1 An illustrative diagram of the CSEM source segments (shown as bold

green line) aligned along the edges of unstructured triangular (tetra-

hedral for 3D) meshes. . . . 188

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F.2 An illustrative diagram of the support of the test function (v

₁

) for a node in Galerkin FE methods. v

₁

is only nonzero within the five triangles connected to node A, or equivalently, the support of v

₁

is the union of the five triangles shown above. v

₁

= 1 at the node A and is zero at all opposite edges of the elements for the node A. Within each element, v

1

linearly changes (assuming linear basis functions are used) from one at the node A to zero at the opposite edge. Therefore, v

1

also changes the same way from node A to node B (or C) at the source

segment (shown as bold green). . . . 189

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List of Tables

2.1 Common types of RBFs. The infinitely smooth RBFs include a shape parameter c > 0. r is the Euclidean distance between two points r

₁

and r

₂

: r = ||r

₁

− r

₂

||

_l₂

. N

⁺

is the space of positive integers. . . . 32 4.1 Differential operators used in eq (4.20). . . . 79 4.2 Calculated values (complex numbers) of normalized ||∇ · A||

2

with

different discretizations. The mode is E-x polarization. The results are for the two frequencies: 0.1 Hz and 10 Hz, respectively. . . . 105 4.3 Computation time for the DTM-1 MT model. The computation run-

ning time includes the time amount of assembling and of solving the

linear system of equations. . . . 113

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Chapter 1 Introduction

1.1 Challenges in constructing realistically com- plex Earth models

1.1.1 Interpretation for realistic geophysical models

Geophysical electromagnetic (EM) methods are extremely sensitive to electrical con-

ductivity in the subsurface of the Earth. As such, EM survey methods have been

the predominant geophysical methods for delineation of conductive or conductor-

associated mineral deposits such as volcanic-associated massive sulphides and ura-

nium deposits (Dyck & West, 1984), and for detection of metal objects such as UXO

(unexploded ordnance) buried in the shallow subsurface (Pellerin, 2002). Surveys

for groundwater, geothermal resources and environmental monitoring also frequently

use EM methods (Everett, 2013). In these applications, the conductivity contrast

between a target and its country rock units is sufficiently high, allowing the target’s

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EM signal to be detectable (Spies & Frischknecht, 1991; Zhdanov, 2009).

Both time-domain and frequency-domain EM survey methods are frequently used in practice. In time-domain EM methods, the EM source signal is generated by inducing a transient electric current within a wire or a loop transmitter, for example, which is then abruptly shut off. The induced secondary EM field from the subsurface can then be observed in the absence of the source signal. In frequency-domain EM methods, however, the source field is continuous and oscillates at a particular period.

The measured EM field in practice is a mix of both source signal and induced signal from the subsurface. A comprehensive summary and explanation of geophysical EM theory and basic survey methods is given by Nabighian (1988, 1991).

The immediate output from an interpretation of the collected EM data for a par- ticular region is a distribution of conductivity, or a model of the physical property conductivity, rather than the actual lithology and rock units. To delineate the litho- logical structures of the subsurface, extra information such as geological knowledge and petrophysical links between the conductivity values and rock types needs to be incorporated into the interpretation. The geophysical data interpretation is itself a quantitative analysis that often consists of forward modelling (computer simulation) and inversion of the surveyed data (Nabighian & Asten, 2002; Oldenburg & Pratt, 2007).

An inversion seeks to recover the conductivity model whose predicted EM re-

sponses fit adequately well the actual field data. The predicted EM responses of a

candidate model are obtained by forward modelling, which is the problem of that, if

the EM survey and configuration parameters and the conductivity distribution are

known, how to find a mathematical solution, for example electric and magnetic fields

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at specific measurement sites, of Maxwell’s equations. One fundamental feature of the inversion is non-uniqueness, meaning that there can be an infinite number of po- tential models whose EM responses fit the observed data (e.g., Parker, 1977). This is precisely the reason why regularized, or constrained, inversions where particular desir- able structures or features of the model are forced to appear are necessary (Tarantola

& Valette, 1982).

In terms of constraints for inversion, either known a priori physical property values or geometries of geological units can be used (McGaughey, 2007). For the former type of information, one example is the air’s conductivity value in a conductivity model with an air layer, which is known and can be treated as a strong constraint. For the latter, examples are surface topography and bathymetry that are often measured by other means. While the physical property information is often straightforward to implement in both forward modelling and inversion, incorporation of known geometry information is more challenging. This is because that in the forward modelling, which is the ‘engine’ of an inversion run, a simple model discretization such as using rectilinear meshes is capable of readily incorporating physical property values, but requires a non-trivial effort to deal with complex geometries. Often, discretizations using unstructured meshes

¹

are needed for representing such geometries, which are ubiquitous in realistic Earth models (Fullagar & Pears, 2007). Indeed, with the increasing power of 3-D geological computer modelling, an integrated geophysical interpretation taking the geological observations such as lithological stratigraphic boundaries and contact surfaces into consideration is needed more than ever (Pears

1An unstructured mesh is a tessellation of the problem domain with irregular connectivity and local topology in the mesh.

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& Chalke, 2016; Leli` evre & Farquharson, 2016).

1.1.2 State-of-the-art numerical modelling methods

In spite of the demand for constructing realistically complex Earth models from geo- physical EM data, 3-D forward modelling and thus inversion algorithms that are capable of doing this have not been reported until recently (e.g., G¨ unther et al., 2006; Newman, 2014; Usui et al., 2017; Jahandari & Farquharson, 2017; Wang et al., 2018). The feasibility of 3-D EM imaging for realistic conductivity models is largely attributed to the adoption of unstructured meshes (e.g., tetrahedral meshes), which permits a relatively easy and efficient discretization of topography and bathymetry (Franke et al., 2007; Nam et al., 2007; Schwarzbach et al., 2011; Ren et al., 2013;

Ansari & Farquharson, 2014; Jahandari & Farquharson, 2015). While the impor- tance of using unstructured meshes in forward modelling was acknowledged several decades ago in academia (Coggon, 1971), a routine use of them has become possible only in recent years when open-source mesh generating software has become avail- able (Shewchuk, 1996; Fabri et al., 2000; Si, 2015). With unstructured meshes being increasingly used, there have been corresponding changes in the development of 3-D numerical modelling algorithms.

For a general inhomogeneous conductivity model, only a numerical solution to Maxwell’s equations can be sought, which is, to date, mostly sought using integral equation (IE) method, finite difference (FD) method, finite volume (FV) method

²

and finite element (FE) method (B¨ orner, 2010). The IE method was intensively studied

2FV method is closely related to FD method, and can be shown to be equivalent to some FD schemes over rectilinear meshes. However, there are also important differences between the two.

Therefore, they are treated as two different numerical methods here.

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from 1970s to 1990s at a time when computer power was very limited (e.g., Jones &

Pascoe, 1972; Hohmann, 1975; Newman et al., 1986). In the IE method, a numerical solution consists of two parts: a primary field which is often computed analytically over a homogeneous or layered Earth model (background model), and a secondary field caused merely by inhomogeneities that are embedded in the background model to complete the 3-D inhomogeneous model. The background model is never discretized, and its EM response can be obtained relatively easily. Only the inhomogeneities are discretized by meshes, either structured or unstructured, and the resultant secondary field is numerically computed. For a single and small localized inhomogeneity, the IE method generates a linear system with a small number of degrees of freedom for solv- ing the secondary field, which can be efficiently solved. However, this linear system has a full and dense matrix, making the method inefficient for modelling complex in- homogeneities. Another challenge of the method is that the background model needs to be some simple, ideal model for which analytical solutions are available. This does however make it difficult to account for complex geometries in a realistic model.

The FD method has been applied to simulate EM fields in engineering problems

for a long time (Yee, 1966; Taflove & Umashankar, 1990). Nevertheless, the method

had not received significant attention in the geophysical EM community until ad-

vanced modern computers were developed in 1990s (Wang & Hohmann, 1993; Mackie

et al., 1993; Newman & Alumbaugh, 1995; Streich, 2009). By discretizing the entire

model domain with meshes (mostly restricted to be rectilinear meshes), FD meth-

ods allow for a more straightforward and general treatment of inhomogeneities in

3-D conductivity models than IE methods. Although the resultant linear system is

typically large, the system is symmetric and highly sparse, and thus can be solved

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with efficiency. One particular aspect of efficiency is its relatively small memory stor- age requirement when the linear system is solved with iterative solvers (Saad, 2003), which was paramount in early 1990s considering the computer power at that time.

Among various FD schemes, the Yee-scheme (Yee, 1966) is particularly favoured since it faithfully reproduces continuous tangential EM fields while at the same time allows for possible jumps in normal EM fields across the elements’ interfaces (material’s discontinuities) in a mesh. In terms of modelling EM responses over geometrically complex models, common FD techniques have severe limitations in dealing with highly irregular geometries since the FD approximation is essentially one dimensional and is only feasible over tensor-product, or orthogonal, grids for 2-D and 3-D problems.

Curvilinear boundaries have to be approximated by staircase-like rectangular cells, or otherwise non-trivial efforts would be needed (Jurgens et al., 1992). Further, a lo- cal refinement of the mesh often propagates to the domain’s boundaries, resulting in excessive degrees of freedom in other regions that are not of interest. The refinement propagation issue can be mitigated by using quadtree (2-D case) or octree (3-D case) meshes, but by doing so extra complexities in computation such as interpolation of unknowns due to unconformity of these meshes have to be introduced (e.g., Haber &

Heldmann, 2007).

For the FE method, the advantages of supporting unstructured meshes and of

generating symmetric and sparse linear systems are both present, making the method

a powerful solution to modelling EM responses over Earth models with complicated

geometries. Like FD methods, the FE method also discretizes the whole problem

domain into subdomains, which are called finite elements. Two important types of

FE approaches are node-based (or scalar) and edge-based (or vector) FE methods. In

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scalar FE methods, unknowns are defined at the nodes in a mesh, whereas in vector FE methods the unknowns are associated with the edges of cells in the mesh. A notable characteristic of the scalar FE method is that the unknown quantity solved for by the numerical method will be forced to be continuous everywhere in the problem domain, in contrast to the vector counterpart where only the tangential component of the unknown quantity is forced to be continuous (Jin, 2014). Consequently, scalar FE techniques are largely restricted to modelling continuous EM field components or EM potentials for inhomogeneous models (Coggon, 1971; Pridmore et al., 1981;

Badea et al., 2001; Li & Key, 2007; Puzyrev et al., 2013).

To date, the FE method appears to be the most suitable modelling method for realistic Earth models with irregular surfaces and topography for example. The reason is that a general treatment of FE approximation of functions over various basic types of subdomains (e.g., triangles and parallelograms in 2-D case, and tetrahedra and prisms in 3-D case) can be used (Brenner & Scott, 2007; Jin, 2014). It almost costs the same amount of effort to implement a FE algorithm using rectilinear meshes as that using completely unstructured meshes. This is in contrast to FD and FV methods, for example. While the FV method can support unstructured meshes, it requires much more effort to do so than that of using a direct FD algorithm with a rectilinear mesh, and it also needs a higher degree of regularity of the unstructured meshes compared to a FE algorithm using the same type of mesh (Jahandari &

Farquharson, 2014, 2015).

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1.1.3 Benefits and challenges in using unstructured meshes

Aside from being more realistic and consistent with geological models, the benefits of using unstructured meshes in understanding the effects of model geometry on simulated EM data have also been consistently demonstrated. For example, G¨ unther et al. (2006) studied artifacts caused by ignoring topography structure in interpreting direct current (DC) resistivity data. Schwarzbach et al. (2011) modelled marine controlled-source EM (CSEM) responses over a bathymetry conductivity model with a seafloor topography constructed from real data, and confirmed the strong influence of topography on the EM fields. The employment of unstructured tetrahedral meshes allowed Jahandari & Farquharson (2015) and Ansari et al. (2017) to synthesize and compare the predicted EM responses of a massive sulphide ore deposit at Voisey’s Bay, Canada, with the actual EM data collected in the field. Another modelling example is the study of Um et al. (2015), where the authors used unstructured meshes in their FE algorithm to investigate the effects of very thin steel casing in a well on borehole- surface CSEM data. In the case of magnetotellurics, Usui (2015) and subsequently Usui et al. (2017) included topography explicitly in the 3-D forward modelling and inversion of real data and demonstrated that the undesired effect of galvanic distortion in interpreting magnetotelluric data can be effectively removed.

With an increasing use of unstructured meshes in tackling 3-D problems, some significant challenges related to mesh generation and FE algorithm implementation have emerged. In order to obtain an accurate numerical solution by the FE method, the mesh used for solving Maxwell’s equations is required to be of sufficient quality.

In the case of FE algorithms, a quality mesh generally means that its cells or ele-

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ments are close to regular polygons or polyhedra. It is understood that the accuracy and efficiency of a FE solution have a strong dependence on the mesh quality, which is often characterized by the ratio of the largest to smallest cell sizes, elongation, dihedral angles and radius-edge ratio of cells, etc, for tetrahedral elements for exam- ple (Du et al., 2009; Ansari & Farquharson, 2014). The process in obtaining a FE numerical solution when using iterative matrix equation solvers, which are memory efficient, can be slow or even fail to converge if the mesh quality is not good enough (Ansari & Farquharson, 2014). Even using direct equation solvers, if without neces- sary quality control in the mesh generation, the resulting mesh can have too many low quality cells which cause numerical interpolation errors in the obtained solution (Schwarzbach et al., 2011). For many real-world Earth models that contain rough and highly irregular changes of contact surfaces and topographies, the requirement of quality meshes thus poses a significant challenge in the mesh generation.

Even with the aid of third-party automated mesh generating software, significant difficulties are still faced. If a quality mesh is generated without guidance, often excessive elements are observed as a result of overmeshing, making the FE-based forward modelling intractable (Nalepa et al., 2016; Nalepa, 2016). An example of meshing a thin layer of hydrocarbon reservoir with possible excessive cells is illustrated in Fig 1.1 (Dunham et al., 2018). For an experienced modeller, some rules of thumb can be used. For example, by deploying denser and smaller elements at the EM transmitters and receivers and in other regions where EM fields are expected to change rapidly, and coarser, larger elements elsewhere, the total number of elements can be made to be affordable for given computing resources (Key & Ovall, 2011).

However, such experience-guided discretization easily becomes non-trivial for large-

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scale models with multiple inhomogeneities where changes of EM fields are harder to predict (Schwarzbach et al., 2011; Nalepa et al., 2016). A practical solution to such manual refinement or coarsening of a mesh is adaptive mesh generation methods (Oden & Prudhomme, 2001). Such adaptive meshing techniques can refine or coarsen some specific parts of an existing mesh based on error estimators constructed from the current numerical solution. As a result, minimum user interference is required in designing a quality mesh without too many excessive cells. One example of adaptive meshing techniques is the goal-oriented adaptive refinement method, which by far has been favoured in 2-D and 3-D EM modelling problems (see, e.g., Franke et al., 2007; Key & Ovall, 2011; Schwarzbach et al., 2011; Ren et al., 2013; Grayver & Burg, 2014).

Nevertheless, adaptive meshing schemes for unstructured elements in 3-D prob- lems are computationally expensive and sometimes even prohibitive. A new mesh resulting from local refinement or coarsening of an existing mesh is obtained either by subdivision or merging of local elements (e.g., Key & Ovall, 2011; Schwarzbach et al., 2011), or by remeshing the whole problem domain (e.g., Ren et al., 2013).

The benefit of the former strategy is that it has good stability and only part of the

original mesh needs to be updated. The difficulty is that complicated mapping of

corresponding elements between the two meshes, and thus updates of degrees of free-

dom, are required. As the topology of the original mesh is changed, noncomformal

elements may also arise, adding additional interpolation computation. In the latter

strategy, the aforementioned issues of mapping are avoided, but the remeshing can be

time-consuming since the whole domain needs to be remeshed at each iteration of re-

finements. Furthermore, the remeshing process suffers from possible breakdown of the

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Figure 1.1. Two different tetrahedral meshes for the same multi-

layer marine hydrocarbon reservoir model. The thin hydrocarbon

layer is shown in red in both meshes. The mesh shown in (a) has

about 1 million more cells than the mesh in (b), with the latter mesh

being generated with better edge length constraints. From Fig 7 in

Dunham et al. (2018).

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mesh generator which can be attributed to various factors such as model complexity, software bugs, memory limits, etc.

To summarize, the FE method in conjunction with unstructured meshes appears to be the most suitable solution among mesh-based methods to modelling 3-D EM re- sponses over real-world Earth models with irregular geometries. However, there have been observed difficulties arising mainly from using quality meshes in the FE method, which potentially hampers its application towards dealing with realistic Earth models.

This observation motivates the author of this thesis to investigate a fundamentally different numerical method, that is, the meshfree method, which does not require a mesh in obtaining a numerical solution, and its applicability in solving the above 3-D EM modelling problems.

1.2 Meshfree discretization in forward modellings

A key feature of meshfree discretizations is that only a cloud of unconnected points

³

are required in deriving a numerical solution to partial differential equations such as Maxwell’s equations. As a consequence, numerical methods that can work with meshfree discretizations in the modelling task have their own distinct features and are termed meshfree methods. This is in contrast to traditional numerical methods that rely on a tessellated mesh (hence, mesh-based methods) in order to acquire a numerical solution with a desired accuracy. It is the mesh reliance of mesh-based methods that requires an Earth model be finely meshed with voxel-like cells or el- ements (Fig 1.2a and Fig 1.2b). However, such fine scale meshes are solely for the

3In the literature of meshfree studies, both ‘node’ and ‘point’ are interchangeably used to refer to the basic element of a meshfree discretization. In this thesis, the word ‘point’ is preferred.

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Classic rectilinear mesh

a) b)

RBF-FD subdomain

Unstructured triangular mesh c)

r

Meshfree points

Figure 1.2. Schematic illustration of model discretizations for 2-D ir- regular geometries using (a) rectilinear mesh with local quadtree re- finements, (b) unstructured triangular mesh and (c) meshfree points.

purpose of geophysical data modelling and are not necessary in describing the ge- ometry of geological structures. In describing a geological model, much rougher 3-D meshes, for example a mesh with only outlines of different rock units, are adequate.

The interior volumes of rock units even need not be meshed. These rough meshes are precisely the outputs and/or inputs from geological interpretations. In this sense, it can be viewed that there are two meshes involved in the context of forward mod- elling geophysical data: one is the mesh describing the geometries of different rock units such as those for a conductor embedded in the resistive subsurface in the half space model (‘model mesh’, Fig 1.3a), and the other one used to carry out forward modelling which is often required to be able to sufficiently sample the function values of the quantity of the forward problem (‘numerical mesh’, Fig 1.3b). Hereinafter, all discussions about meshes will refer to the latter case.

A meshfree discretization serves the same purpose as that of a tessellated mesh

(Fig 1.2c) used for a good numerical solution. For this reason, the discretization

is also required to be fine enough in most scenarios (Fig 1.3c). However, in the

case of a meshfree discretization, there are now important advantages regarding the

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(a) (b) (c) Figure 1.3. Different spatial discretizations of an Earth model in mesh-based and meshfree numerical methods: (a) a mesh describ- ing the example Earth model which is shown as the cross section of a conductor (in red) embedded in the half space (in grey); (b) an example mesh (unstructured tetrahedra) used for solving the equa- tions if using mesh-based methods; (c) a cloud of points used for solving the equations when using meshfree methods.

mesh generation and mesh refinement issues over traditional mesh-based numerical methods:

• The generation of a set of unconnected points, even with comparable regularity constraints of a fine mesh, is more straightforward and requires less effort.

• The generation of quality points is believed to be more robust than the gen- eration of quality meshes (Du et al., 2002; Fornberg & Flyer, 2015b; Slak &

Kosec, 2019), allowing for a more robust adaptive refining (‘remeshing’) than for mesh-based methods.

• Adaptive refining and/or coarsening of a meshfree discretization is more compu-

tationally efficient than the similar process in mesh-based numerical methods,

since an addition or deletion of some points will not affect the rest of the points

(Duarte & Oden, 1996b; Rabczuk & Belytschko, 2005). This means that there

will be no unconformity issues between the old degrees of freedom and the new

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ones.

These advantages provide the basis for overcoming the aforementioned challenges in modelling EM data over complex Earth models when using mesh-based methods such as FE. Meshfree numerical methods for data simulation were originally developed in research fields other than geophysical applications. The investigation in this thesis is therefore focused on how, given a meshfree point discretization, can an appropriate formulation of Maxwell’s equations be solved using such a discretization.

1.3 Thesis overview

In the following, Chapter 2 firstly presents a review of meshfree numerical methods.

Basic meshfree function approximation methods as well as state-of-the-art meshfree numerical methods are introduced. The meshfree method investigated in this thesis is also elaborated with implementation details and justifications for it to be used for geophysical data modelling. In Chapter 3, with the background problem being that of numerically modelling gravity data, a convergence analysis of the meshfree method based on numerical evidence is presented, which is followed by the investigation of how to determine the stencil size (the number of points in a meshfree subdomain) in implementing the meshfree method. A comparison study using the meshfree ap- proach and a more traditional finite element algorithm is also provided. This part of work has been published as a peer-reviewed paper (Long & Farquharson, 2019a).

Chapter 4 presents the feasibility and applicability study of the proposed meshfree

method in numerically simulating 3-D magnetotelluric data, a commonly used EM

survey approach that uses passive EM source signals. The feasibility of using the

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meshfree method in the frame of EM potential equations is theoretically analyzed first and then demonstrated with numerical results. The results of this chapter have been published in a separate paper (Long & Farquharson, 2019c). In Chapter 5, the mixed meshfree method proposed in modelling magnetotelluric data is further extended in order to solve the forward modelling of controlled-source EM responses over a general 3-D conductivity model. The incorporation of singular EM source func- tions in the meshfree method is investigated and numerically tested. Part of the work in this chapter has been summarized in previous SEG Expanded Abstracts (Long &

Farquharson, 2017, 2019b). Finally, Chapter 6 and 7 present further discussions and

conclusions, respectively, of the above studies.

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Chapter 2 A review of meshfree numerical methods

In this chapter, function approximation in a mathematical background is presented first, which is followed by a review of meshfree numerical methods for solving general partial differential equations in the literature. The meshfree method investigated in this thesis is reviewed with more details regarding its development history and algorithm implementation, along with the justifications of employing it in simulating geophysical data.

2.1 Meshfree function approximation

Meshfree function approximation is fundamental to meshfree numerical methods for solving partial differential equations (PDEs). The problem of meshfree func- tion approximation can be stated as: given n discrete data sites with positions r

i

(i = 1, . . . , n) and their associated function values {f

i

}

ⁿ_i=1

, find an approximant ˆ f(r)

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to the unknown function f, which is assumed to be sufficiently smooth, such that

|| f(r) ˆ − f (r)||

_p

, r ∈ Ω ⊂ R

³

, f , f ˆ ∈ R , (2.1)

is minimized over the problem domain Ω. || · ||

_p

is a chosen norm, and R

³

is the 3-D space of real values. Ω is the support of f and is also called a meshfree subdomain when only a set of local points are employed to construct function approximation (see Fig 2.1). Eq (2.1) states a general measure of the quality of the approximation. To obtain a good approximation, the approximate ˆ f (r) is often expanded using known and simple functions.

The assumption of a sufficient smoothness of the unknown function allows a gen- eral treatment of the approximation problem and systematic developments of math- ematical theories (e.g., error analysis, see Buhmann, 2003; Fasshauer, 2007). With this assumption, the approximant, ˆ f (r), can be expanded as a linear combination of some basis functions ϕ

f(r) = ˆ

M

∑

i=1

ϕ

_i

(r) · c

_i

, (2.2)

where c

i

are coefficients to be determined, and M is the number of terms of basis functions. Depending on the choice of types of basis functions, M may or may not be equal to n which is the number of data sites.

In the context of meshfree approximation where the data sites are discrete points

in 3-D space, two types of basis functions are commonly used: monomial functions

and radial functions. Other rational functions are possible in principle, but they

are not as widely used as these two types (Fasshauer, 2007). The approximation

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10(N₁)

20(N2) 7

21 35 34

2

support point N₂ support point N₁

inﬂuence domains (subdomains)

Figure 2.1. Schematic illustration of meshfree subdomains consisting of scattered points. A set of local points within a neighbourhood (outlined by dotted circles) of a point (denoted as support point) comprises the subdomain for that point.

based on monomial functions is termed moving least squares (MLS) method, and the counterpart based on radial basis functions is termed radial basis function-based method (Fasshauer, 2007). Note the use of monomial functions in approximation over meshed subdomains is the foundation of the mesh-based numerical methods such as finite element and classical grid-based finite difference methods (Fig 2.2 (a) and (b)).

In the method of MLS, a complete set of monomial functions is typically used for accuracy. For example, a complete set of linear monomials in 3-D space is [1, x, y, z]

with M = 4, and a complete set of quadratic monomials is [1, x, y, z, xy, yz, xz, x

²

, y

²

, z

²

] with M = 10. The approximant ˆ f constructed using MLS is determined by minimiz- ing the functional

J =

n

∑

k=1

w(r, r

_k

) (

_M

∑

i=1

ϕ

_i

(r) · c

_i

− f

_k

)

²

, (2.3)

where c

_i

are the coefficients to be determined, r

_k

is the position of kth point in a

subdomain, and w(r, r

_k

) is a weight function which is predetermined and is chosen

such that w > 0. In order to obtain a unique solution to eq (2.3), n ⩾ M is required

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support node

inﬂuence domain (subdomain)

r

1

2 3

unstructured mesh

b) c)

1

2 3

f

1

f

2

f

3

a)

grid-based ﬁnite diﬀerence

Figure 2.2. Schematic illustration of different discretizations in mesh-based and meshfree approximation methods. Shown above are (a) a grid-based finite difference stencil, (b) an unstructured mesh and (c) two meshfree subdomains.

to hold. In practice, n > M is often the case and c

_i

are solved in a least squares manner (see Nguyen et al., 2008, and references therein). The weight function is required here so that the linear system of equations from least squares averaging has a nonsingular matrix for high dimensional (2-D and 3-D) data sites, which makes MLS feasible for multivariate function approximation, or equivalently, for high dimensional meshfree function approximation. However, in 2-D and 3-D situations, the data sites, or meshfree points in a subdomain are still required to possess a certain degree of regularity such as not being collinear points (Fig 2.3), regardless of the choice of w.

In the method of approximation by radial basis functions (RBFs), which are ex-

plained with more details in Sections 2.4 and 2.4.1, ϕ

_i

(r) are translations of a chosen

RBF at the n points. Since many RBFs that are used in practical meshfree function

approximation are positive definite functions, standard Lagrange-type interpolation

conditions ( ˆ f (r

_i

) = f

_i

= f (r

_i

), i = 1, · · · , n) can be employed to determine c

_i

in

eq (2.2). In this case, n = M in each subdomain.

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Equidistant points

(a) (b)

Random points

(c)

collinear points

Figure 2.3. Some point arrangements in a 2-D meshfree subdomain:

(a) regular, equidistant points; (b) randomly distributed points and (c) collinear points. The red point is the support point.

2.2 Meshfree methods for solving PDEs

Different meshfree numerical methods have been developed to solve PDEs arising from various applications. Although early applications of meshfree methods were pri- marily focused on fluid and solid mechanics and astrophysics, applications in many other fields have been reported (see the reviews by Belytschko et al., 1996; Li & Liu, 2002; Nguyen et al., 2008; Chen et al., 2017). Over the years, there have been nu- merous meshfree PDE-solving approaches proposed which are often named by the proposers’ practical applications, causing a remarkable inconsistency in the nomen- clature of meshfree numerical methods (see, e.g., the various names listed in Chen et al., 2017). This is sometimes inconvenient and even misleading for researchers and practitioners to understand the techniques in those approaches. To alleviate this issue, an explanation of the essence of those meshfree methods is presented here.

Similar to mesh-based numerical methods, meshfree methods are generally cate-

gorized as two groups from the point of view of deriving a numerical solution: strong

form-based methods (SFMs) and weak form-based methods (WFMs). In the process

of seeking a numerical solution to PDEs (e.g., Df = g, g is a known function) over a

bounded domain, the action of a differential operator D on the unknown function f,

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Df , will be approximated as the result of a linear combination of a certain number (e.g., k) of local function values

Df ≈

k

∑

j=1

c

_j

f

_j

. (2.4)

In SFMs, the coefficients c

_j

in eq (2.4) are obtained by satisfying the above approx- imation at discrete data sites (meshfree points here). In WFMs, however, the PDE under consideration is satisfied in an averaged sense (thus, the ‘weak’ form)

∫

Ω^′

v · (Df − g) dV = 0, Ω

^′

⊂ R

³

, (2.5)

where v is often referred to as test function or weight function, and Ω

^′

is the problem domain.

It can be observed from eqs (2.4) and (2.5) that in either type of the numerical methods, approximation of the unknown function f needs to be carried out before a numerical solution can be found, and there are a number of meshfree function approx- imation approaches available in addition to the two widely used ones mentioned in the previous section. In the case of WFMs, aside from different ways of approximating f, there are different methods to partition the problem domain into subdomains (sub- domains for function approximation and for weak form integration may be different).

There are also different ways to choose the test function, for which two commonly used approaches are Galerkin (also known as Bubnov-Galerkin or standard Galerkin) method and Petrov-Galerkin method (Atluri & Zhu, 1998; Gockenbach, 2006).

Differing in these aspects in finding a numerical solution, some notable strong

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form-based meshfree methods include smoothed particle hydrodynamics (Gingold &

Monaghan, 1977), generalized finite differences (Liszka & Orkisz, 1980), the vortex

method (Leonard, 1985), finite-volume particle-in-cell method (Munz et al., 1999),

meshless collocation method (Kansa, 1990a) and RBF-based finite difference (RBF-

FD, Tolstykh & Shirobokov, 2003). Similarly, some notable weak form-based mesh-

free methods are diffuse element method (Nayroles et al., 1992), element free Galerkin

method (EFG, Belytschko et al., 1994), reproducing kernel particle method (Liu et al.,

1995), h-p cloud method (Duarte & Oden, 1996a), partition of unity method (Me-

lenk & Babuˇska, 1996), free mesh method (Yagawa & Yamada, 1996), meshless local

Petrov-Galerkin method (MLPG, Atluri & Zhu, 1998), and radial point interpola-

tion method (RPIM, Wang & Liu, 2002a). In the past two decades, improvements

to the aforementioned methods have been developed that are devoted to enhancing

particular aspects in computer implementations such as efficiency and convergence

rate (Chen et al., 2017). One important example of these improvements is how to

accurately and efficiently carry out volume or surface integration over meshfree sub-

domains in WFMs. This is because when using meshfree function approximation

techniques, the resultant shape functions are often high-order rational functions, as

opposed to low-order polynomials that typically appear in mesh-based methods (e.g.,

finite element and integral equation methods), and as such the integrals in eq (2.5)

have to be carried out numerically instead of analytically.

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2.3 Applications in geophysics

Applications of meshfree methods in geophysics were reported as early as the 1970s (Hardy, 1971). However, meshfree methods are still not widely known to the geo- physical community. As mentioned in the Introduction of the thesis, a major benefit of using meshfree numerical methods is that irregular geometries can be relatively easily represented. Another advantage using meshfree methods is that high order accuracy and/or convergence rate can be achieved by a discretization with fewer de- grees of freedom than that using traditional mesh-based methods. The latter is due to the high order smoothness of the meshfree approximant or interpolant. To date, there have been increasing applications of meshfree methods in forward modelling geophysical data. For example, Jia & Hu (2006) investigated the numerical accu- racy of applying EFG to simulate 2-D time-domain seismic wave fields. Martin et al.

(2015) demonstrated that using RBF-FD with multiquadric RBFs can lead to a high- order h convergence in modelling 2-D seismic wave propagation. Similarly, Takekawa et al. (2015) proposed to use a polynomial-based meshfree FD approach to solve the problem of frequency-domain elastic wave modelling, which is further studied by Takekawa & Mikada (2016, 2018). Li et al. (2017) studied time-space-domain elastic wave modelling using RBF-FD for greater temporal accuracy, which uses the same RBFs as in Martin et al. (2015). For the modelling of EM data, Wittke & Tezkan (2014) presented an application of MLPG for simulating 2-D magnetotelluric (MT) responses of inhomogeneous conductivity models. The same MT numerical problem was also studied by Li et al. (2015) using an implementation of EFG and by Ji et al.

Forward modelling 3-D geophysical electromagnetic field data with meshfree methods